Question

1. One fair coin and two unfair coins where heads is 5 times as likely as tails are put into a bag. One coin is drawn at random and then flipped twice. If at least one of the flips was tails, what is the probability an unfair coin was flipped?

2. Every day, Janet either takes the bus or drives her car to work. She drives her car 30% of the time. When she drives her car, she packs her lunch 70% of the time. When she takes the bus she packs her lunch 20% of the time.

   (a) Given that Janet rode the bus, what is the probability that she did not pack her lunch?   (b) Given that Janet did not pack her lunch, what is the probability that she rode the bus

Answer 1

1)

P(at least one tails) =P(fair coin)*P(at least one tails| fair coin)+P(unfair coin)*P(at least one tails| unfair coin)

=(1/3)*(1-(1/2)*(1/2))+(2/3)*(1-(5/6)*(5/6))

=49/108

therefore P( unfair coin given at least one tails)

=P(unfair coin)*P(at least one tails| unfair coin)/P(at least one tails)

=(2/3)*(1-(5/6)*(5/6))/(49/108)

=22/49

2)

P(not pack the lunch | rode the bus )=1-P(pack the lunch |rode the bus ) =1-0.20 =0.80

b)

P(rode the bus|not pack the lunch) =P(rode the bus and not pack the lunch)/P(not pack the lunch)

=(1-0.30)*(1-0.2)/((1-0.30)*(1-0.2)+0.3*(1-0.70))=0.8615